3,370 research outputs found
On the number of k-dominating independent sets
We study the existence and the number of -dominating independent sets in
certain graph families. While the case namely the case of maximal
independent sets - which is originated from Erd\H{o}s and Moser - is widely
investigated, much less is known in general. In this paper we settle the
question for trees and prove that the maximum number of -dominating
independent sets in -vertex graphs is between and
if , moreover the maximum number of
-dominating independent sets in -vertex graphs is between
and . Graph constructions containing a large number of
-dominating independent sets are coming from product graphs, complete
bipartite graphs and with finite geometries. The product graph construction is
associated with the number of certain MDS codes.Comment: 13 page
The still-Life density problem and its generalizations
A "still Life" is a subset S of the square lattice Z^2 fixed under the
transition rule of Conway's Game of Life, i.e. a subset satisfying the
following three conditions:
1. No element of Z^2-S has exactly three neighbors in S;
2. Every element of S has at least two neighbors in S;
3. Every element of S has at most three neighbors in S.
Here a ``neighbor'' of any x \in Z^2 is one of the eight lattice points
closest to x other than x itself. The "still-Life conjecture" is the assertion
that a still Life cannot have density greater than 1/2 (a bound easily
attained, for instance by {(x,y): x is even}). We prove this conjecture,
showing that in fact condition 3 alone ensures that S has density at most 1/2.
We then consider variations of the problem such as changing the number of
allowed neighbors or the definition of neighborhoods; using a variety of
methods we find some partial results and many new open problems and
conjectures.Comment: 29 pages, including many figures drawn as LaTeX "pictures
Minimizing the regularity of maximal regular antichains of 2- and 3-sets
Let be a natural number. We study the problem to find the
smallest such that there is a family of 2-subsets and
3-subsets of with the following properties: (1)
is an antichain, i.e. no member of is a subset of
any other member of , (2) is maximal, i.e. for every
there is an with or , and (3) is -regular, i.e. every point
is contained in exactly members of . We prove lower
bounds on , and we describe constructions for regular maximal antichains
with small regularity.Comment: 7 pages, updated reference
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